I made mathematical model of periodical behavior and internal state of quadruped leg animal and paramecium. The gait transition of quadruped leg animal is reproduced by assuming damped oscillator. This damped oscillators are made by physical interaction. By this results the robot which can change their gait pattern with only physicalinteraction is made in Ishiguro Akio's Laboratory (Tohoku Univ.) We made mathematical model for the experiment in which paramecium can remember vessel shape. Now the mathematical model is compared with real paramecium by the experiment in Toshiyuki Nakagaki's Laboratory(Hakodate Future Univ.)